How do I know when a user clicks a Like button?
If you are using the XFBML version of the button, you can subscribe to the 'edge.create' event through FB.Event.subscribe.

When you include the Like button as an iFrame directly, there's no way for the event from Facebook's servers to be propagated back to your javascript code - the Facebook Javascript SDK handles this communication when using the XFBML version of the Like button